A Multivariate Model of Achievement in Geometry

ABSTRACT

Previous studies have shown that several key variables influence student achievement in geometry, but no research has been conducted to determine how these variables interact. A model of achievement in geometry was tested on a sample of 102 high school students. Structural equation modeling was used to test hypothesized relationships among variables linked to successful problem solving in geometry. These variables, including motivation, achievement emotions, pictorial representation, and categorization skills, were examined for their influence on geometry achievement. Results indicated that the model fit well. Achievement emotions, specifically boredom and enjoyment, had a significant influence on student motivation. Student motivation influenced students’ use of pictorial representations and achievement. Pictorial representation also directly influenced achievement. Categorization skills had a significant influence on pictorial representations and student achievement. The implications of these findings for geometry instruction and for future research are discussed.

Keywords:

• achievement emotions
• geometry
• mathematics
• motivation
• multivariate statistics
• problem solving

APPENDIX A

Geometry Motivation Questionnaire

In order to better understand what you think and how you feel about your high school geometry courses, please respond to each of the following statements from the perspective of: “When I am in a high school geometry course…”

1. I enjoy learning the geometry.

   □ Never □ Rarely □ Sometimes□ Usually□ Always

2. The geometry I learn relates to my personal goals.

   □ Never □ Rarely □ Sometimes □ Usually □ Always

3. I like to do better than the other students on the geometry tests.

   □ Never □ Rarely □ Sometimes□ Usually□ Always

4. I am nervous about how I will do on the geometry tests.

   □ Never □ Rarely □ Sometimes□ Usually□ Always

5. If I am having trouble learning the geometry, I try to figure out why.

   □ Never □ Rarely □ Sometimes□ Usually□ Always

6. I become anxious when it is time to take a geometry test.

   □ Never □ Rarely □ Sometimes□ Usually□ Always

7. Earning a good geometry grade is important to me.

   □ Never □ Rarely □ Sometimes□ Usually□ Always

8. I put enough effort into learning the geometry.

   □ Never □ Rarely □ Sometimes□ Usually□ Always

9. I use strategies that ensure I learn the geometry well.

   □ Never □ Rarely □ Sometimes□ Usually□ Always

10. I think about how learning the geometry can help me get a good job.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

11. I think about how the geometry I learn will be helpful to me.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

12. I expect to do as well as or better than other students in the geometry course.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

13. I worry about failing the geometry tests.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

14. I am concerned that the other students are better in geometry.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

15. I think about how my geometry grade will affect my overall grade point average.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

16. The geometry I learn is more important to me than the grade I receive.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

17. I think about how learning the geometry can help my career.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

18. I hate taking the geometry tests.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

19. I think about how I will use the geometry I learn.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

20. It is my fault, if I do not understand the geometry.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

21. I am confident I will do well on the geometry assignments and projects.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

22. I find learning the geometry interesting.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

23. The geometry I learn is relevant to my life.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

24. I believe I can master the knowledge and skills in the geometry course.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

25. The geometry I learn has practical value for me.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

26. I prepare well for the geometry tests and quizzes.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

27. I like geometry that challenges me.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

28. I am confident I will do well on the geometry tests.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

29. I believe I can earn a grade of “A” in the geometry course.

    □ Never □ Rarely □ Sometimes□ Usually□ Always

30. Understanding the geometry gives me a sense of accomplishment.

    □ Never □ Rarely □ Sometimes □Usually □Always

APPENDIX B

Achievement Emotions in Geometry

Studying geometry can induce different feelings. This questionnaire refers to emotions you may experience when studying geometry. Before answering the questions below, please recall some typical situations of studying geometry which you have experienced during the course of your studies.

1. I look forward to studying geometry.

   ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

2. I enjoy the challenge of learning the geometry material.

   ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

3. I enjoy acquiring new geometry knowledge.

   ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

4. I enjoy dealing with the geometry material.

   ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

5. Reflecting on my progress in my geometry coursework makes me happy.

   ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

6. I study geometry more than required because I enjoy it so much.

   ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

7. I am so happy about the progress I made that I am motivated to continue studying geometry.

   ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

8. Certain geometry subjects are so enjoyable that I am motivated to do extra readings about them.

   ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

9. When my geometry studies are going well, it gives me a rush.

   ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

10. I get physically excited when my geometry studies are going well.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

11. The geometry material bores me to death.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

12. Studying for my geometry class bores me.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

13. Studying geometry is dull and monotonous.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

14. While studying this boring geometry material, I spend my time thinking of how time stands still.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

15. Geometry is so boring that I find myself daydreaming.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

16. I find my mind wandering while I study geometry.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

17. Because I’m bored I have no desire to learn geometry.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

18. I would rather put off this boring geometry work till tomorrow.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

19. Because I’m bored I get tired sitting at my desk and studying geometry.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

20. Geometry bores me so much that I feel depleted.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

21. While studying geometry I seem to drift off because it's so boring.

    ○ Strongly Disagree ○ Disagree ○ Undecided ○ Agree ○ Strongly Agree

APPENDIX C

Geometry Problems Used to Assess Pictorial Representation

Circle your answer choice for each problem below.

1. A 15-foot tree makes an 8-foot shadow. What is the sine of the angle between the ground and the ray of sunlight creating the shadow?

A. 8/15 B. 15/8

C. 8/17 D. 15/17

2. A lighthouse on the shore of Point Loma stands 75 feet high. From the observation deck on top of the lighthouse a watchman determines that a ship is located at 17° angle of depression from the horizontal. How far is the boat from the lighthouse? Round your answer to the nearest foot.

Table

A. 23 feet B. 245 feet

C. 72 feet D. 257 feet

3. and are two tangent segments to the same circle from point E with points of tangency A and B respectively. AE = 5 km, what is BE?

A. 2.5 km B. 5 km

C. 10 km D. 15 km

4. An equilateral triangle is inscribed in a circle. What is the measure of the central angle formed by two radii going to two adjacent vertices of the triangle?

A. 30° B. 60°

C. 90° D. 120°

5. A tangent and a chord intersect to form a 75° angle. What is the measure of the major arc formed by the chord?

A. 285° B. 105°

C. 210° D. 185°

6. What is the perimeter of a 45–45 right triangle with a hypotenuse length of 28 feet?

A. 56 ft B. 64.5 ft

C. 67.6 ft D. 84 ft

7. Beatriz entered her collie in a dog show. During the main event, she will walk in a 129° arc in front of the judges. If the arc were to continue in a circle, its radius would be 3.5 feet. What is the distance Beatriz and her dog will need to walk (find the arc length)? Round your answer to the nearest hundredth. Use Π = 3.14.

A. 7.88 feet B. 15.75 feet

C. 3.50 feet D. 7.00 feet

8. What is the length of an altitude of an equilateral triangle with sides of length 24 in. rounded to the nearest tenth?

A. 41.6 in. B. 20.8 in.

C. 33.9 in. D. 17.0 in.

9. You are standing 16 feet from a circular swimming pool. The distance from you to a point of tangency on the pool is 32 feet. What is the radius of the swimming pool?

A. 24 ft B. 22.6 ft

C. 64 ft D. 48 ft

10. Two tangents form an angle outside of a circle such that the measure of one of the arcs formed is 138°, what is the measure of the angle?

A. 138° B. 111°

C. 42° D. 84°

11. If , then cos M = ?

A. B.

C. D.

12. A band must know the dimensions of the stage where they are having a concert. The area of the rectangular stage is 85 ft² and the width is x. The length of the stage is (4x − 3) feet. What is the length of the stage?

A. 17 ft B. 50 ft

C. 21 ft D. 43 ft

APPENDIX B

Pictorial Representation Scoring Rubric

Table

APPENDIX E

Categorization Tasks

Cut out each problem below. Without having to solve the problems, group the problems in pairs however you choose. Glue or tape the pairs in the table provided, and describe why you grouped the pairs in the way you did.

APPENDIX F

Categorization Activity Rubric

Notes

¹Properties of quadrilaterals is a prerequisite concept.

²Eight students in the study scored a zero on one of the unit tests due to an absence. These students were given the opportunity to take the missed test any time before the end of the semester, but did not do so. With the eight students dropped from analysis, results (correlation matrix, model relations, fit indices, and path values) remained the same with only very minor differences in path and correlation values. This was also true when mean substitution was used to replace the eight zero scores.

Additional information

Notes on contributors

MarLynn Bailey

MarLynn Bailey and Gita Taasoobshirazi are researchers in the Department of Educational Psychology at Kennesaw State University.

Gita Taasoobshirazi

MarLynn Bailey and Gita Taasoobshirazi are researchers in the Department of Educational Psychology at Kennesaw State University.

Martha Carr

Martha Carr is a researcher in the Department of Educational Psychology at the University of Georgia.